The first order properties are linked with Ehrenfeucht-Fraïssé games [1]}.
It is well-known that two structures satisfy the same first order sentences with quantifier depth \(d\)  if and only if
the duplicator has a winning strategy for the \(d\) -round Ehrenfeucht-Fraïssé game played on two structures,
in our settings, on two 2-edge-colored rooted marked \(k\) -trees.
The \(d\) -round Ehrenfeucht-Fraïssé game is played by two players, called the spoiler and the duplicator.
In the \(i\) -th round, the spoiler chooses a vertex of one of the \(k\) -trees (the spoiler can choose a different \(k\) -tree
in different rounds) and places the \(i\) -th pebble on that vertex.
The duplicator responds with placing the \(i\) -th pebble on a vertex of the other \(k\) -tree.
At the end of the game,
the duplicator wins
if the mapping between the subtrees induced by the vertices with the pebbles that
maps the vertex with the \(i\) -pebble in the first \(k\) -tree to the vertex with the \(i\) -pebble in the other \(k\) -tree
is an isomorphism preserving the \(j\) -parent relations, \(j\in [k]\) , and the edge colors.
If two 2-edge-colored rooted marked \(k\) -trees satisfy the same \(d\) -sentences if and only if
the duplicator has a winning strategy for the \(d\) -round Ehrenfeucht-Fraïssé game.
